Background

In this example, we will show how to use the Emulator to prepare ordered ground states in Rydberg systems. The example is based on experimental works in 1D system, and 2D system.

Due to the strong Rydberg interactions, only one Rydberg excitation is allowed within the so-called blocakde radius Blockade. With positive detunings, more Rydberg excitations, however, are favored to lower the energies of the ground states. With the interplay of these two mechanisms, different ordered states are supported depending on the strength of blockade radius and the detuning. For example, in 1D there exists the $Z_N$ ordered states, and in 2D there are the checkboard phase, star phase, and even pure quantum phase (straited phase).

The adibatic process works as follows: One starts from all the atoms being in the ground state, which is also the ground states of many-body Hamiltonian with large negative detuning. Then the Rabi freuency $\Omega$ is turned on, and the detuning strength is ramped up from large negative value to postive values. If such process is slow enough, the quantum state the system stays close to the ground state of the instantaneous Hamiltonian. At the end of the process, one arrives a target Hamiltonian, and correspondingly, the prepared state is approximately the ground state for the final Hamiltonian.

We start by importing required libraries

using EaRyd
using CairoMakie
using EaRydPlots
using KrylovKit
using SparseArrays

Ground state properties

To get a sense of the quantum phase transition, let us first calculate the ground states' properties for Hamiltonians with different parameters. We will focus on ground state of 1D chain for simplicity. We consider a chain with 9 atoms, where nearby atoms are seperated by a distance of 5.72 $\mu m$. Such a system is generated by the following codes

nsites = 9
atoms = generate_sites(ChainLattice(), nsites, scale=5.72)

We fix the Rabi frequency to be $Ω = 2π * 4$, and study the ground state as a function of detuning $Δ$.

Ω = 2π * 4
Δ_step = 30
Δ = LinRange(-10*2π , 10*2π , Δ_step)

30-element LinRange{Float64, Int64}:
 -62.8319,-58.4986,-54.1654,-49.8322,…,49.8322,54.1654,58.4986,62.8319

We compute the Rydberg density profile for each parameter of $\Delta$.

density_g = zeros(Δ_step, nsites)

for ii=1: Δ_step

    h_ii = rydberg_h(atoms; Δ=Δ[ii], Ω)
    h_m = SparseMatrixCSC(h_ii)
    vals, vecs, info = KrylovKit.eigsolve(h_m,  1, :SR)
    g_state = ArrayReg(vecs[1])

    for jj in 1:nsites
        density_g[ii, jj] = real(expect(put(nsites, jj=>Op.n), g_state))
    end

end

To compare, we first plot the density profile when $\Delta= -10*2π$,

lines(1:nsites, density_g[1, :])

We can see that the Rydberg densities in this case is close to 0 for all sites. In contrast, for $\Delta= -10*2π$, the density shows clear $Z_2$ ordered profile

lines(1:nsites, density_g[20, :])

More generally, we can plot an order parameter as a function of $\Delta$ to clearly see the onset of phase transition. Such a parameter is defined by the difference of Rydberg densities in even and odd sites.

order_para = map(1: Δ_step) do ii

    sum(density_g[ii, 1:2:8])-  sum(density_g[ii, 2:2:8])

end


lines(collect(Δ)/2π, order_para)

From the density profile of ground states and the change of order parameter, we are convinced that there is indeed a phase transition taking place during the change of $\Delta$. Below we show that by slowly changing the parameters in the Hamiltonian, we can follow the trajectory of the ground states and adibatically evolve from all atoms being in state 0 to the $Z_2$ ordered state.

Ordered state prepration in 1D

We first specify the adiabatic pulse sequence for Rabi frequency by using the built-in waveform function piecewise_linear

total_time = 3.0
Ω_max = 2π * 4
Ω = piecewise_linear(clocks=[0.0, 0.1, 2.1, 2.2, total_time], values=[0.0, Ω_max, Ω_max, 0, 0])
draw(Ω)


#Δ1 = piecewise_linear(clocks=[0.0, 0.6, 2.1, 3], values=[-10*2π, -10*2π, 10*2π, 10* 2π])

The detuning sequence could also be created in a similar way

U1 = -2π * 10
U2 = 2π * 10
Δ = piecewise_linear(clocks=[0.0, 0.6, 2.1, total_time], values=[U1, U1, U2, U2])
draw(Δ)

We then generate the positions of a 1D atomic chain by using the function generate_sites

nsites = 9
atoms = generate_sites(ChainLattice(), nsites, scale=5.72)

Note that we have specify the nearest-neighbor atoms being seperated by 5.72 $\mu m$ in order to prepare a $Z_2$ ordered state. With the waveforms and atomic coordinate specified, the time-dependent Hamiltonian can be simply generated by the following

h = rydberg_h(atoms; Δ, Ω)

Hamiltonian
  Term 1
   ∑(n=1:9) Waveform(_, 3.0)(t)/2 σ^x

  Term 2
   -∑(n=1:9) Waveform(_, 3.0)(t) n

  Term 3
   ∑(n=1:9) 5.42e6/|r_i - r_j|^6 n_i n_j

We then specify all atoms to be in ground state initially, and set up the emulation problem by choosing the ODE solver

reg = zero_state(9)
prob = SchrodingerProblem(reg, total_time, h; dt=1e-5, adaptive=false)
integrator = init(prob, Vern8())

t: 0.0
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We measure the Rydberg density for each site and time step

densities = []
for _ in integrator
    push!(densities, [expect(put(nsites, i=>Op.n), reg) for i in 1:nsites])
end
D = hcat(densities...)

9×300000 Matrix{ComplexF64}:
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3.11485e-10+0.0im  3.35666e-10+0.0im  3.61228e-10+0.0im  3.88224e-10+0.0im  4.16705e-10+0.0im  4.46724e-10+0.0im  4.78338e-10+0.0im  5.116e-10+0.0im  5.46567e-10+0.0im  5.83297e-10+0.0im  6.21848e-10+0.0im  6.62278e-10+0.0im  7.04649e-10+0.0im  7.49021e-10+0.0im  7.95456e-10+0.0im  8.44017e-10+0.0im  8.94769e-10+0.0im  9.47775e-10+0.0im  1.0031e-9+0.0im  1.06082e-9+0.0im  1.12099e-9+0.0im  1.18368e-9+0.0im  1.24897e-9+0.0im  1.31692e-9+0.0im  1.38761e-9+0.0im  1.4611e-9+0.0im  1.53748e-9+0.0im  1.61681e-9+0.0im  1.69917e-9+0.0im  1.78464e-9+0.0im  1.8733e-9+0.0im  1.96521e-9+0.0im  2.06047e-9+0.0im  2.15915e-9+0.0im  2.26133e-9+0.0im  2.3671e-9+0.0im  2.47653e-9+0.0im  2.58972e-9+0.0im  2.70674e-9+0.0im  2.82768e-9+0.0im  2.95263e-9+0.0im  3.08168e-9+0.0im  3.21491e-9+0.0im  3.35241e-9+0.0im  3.49427e-9+0.0im  3.6406e-9+0.0im  3.79147e-9+0.0im  3.94698e-9+0.0im  4.10722e-9+0.0im  4.2723e-9+0.0im  4.4423e-9+0.0im  4.61732e-9+0.0im  4.79747e-9+0.0im  4.98283e-9+0.0im  5.17352e-9+0.0im  5.36962e-9+0.0im  5.57125e-9+0.0im  5.7785e-9+0.0im  5.99148e-9+0.0im  6.2103e-9+0.0im  6.43505e-9+0.0im  6.66585e-9+0.0im  6.9028e-9+0.0im  7.14601e-9+0.0im  7.39559e-9+0.0im  7.65165e-9+0.0im  7.9143e-9+0.0im  8.18366e-9+0.0im  8.45983e-9+0.0im  8.74294e-9+0.0im  9.03308e-9+0.0im  9.33039e-9+0.0im  9.63498e-9+0.0im  9.94697e-9+0.0im  1.02665e-8+0.0im  1.05936e-8+0.0im  1.09285e-8+0.0im  1.12713e-8+0.0im  1.1622e-8+0.0im  1.19809e-8+0.0im  1.2348e-8+0.0im  1.27235e-8+0.0im  1.31075e-8+0.0im  1.35002e-8+0.0im  1.39015e-8+0.0im  1.43118e-8+0.0im  1.47311e-8+0.0im  1.51595e-8+0.0im  1.55972e-8+0.0im  1.60443e-8+0.0im  1.6501e-8+0.0im  1.69673e-8+0.0im  1.74434e-8+0.0im  1.79295e-8+0.0im  1.84257e-8+0.0im  1.89321e-8+0.0im  1.94488e-8+0.0im  1.99761e-8+0.0im  2.0514e-8+0.0im  2.10627e-8+0.0im  2.16223e-8+0.0im  2.2193e-8+0.0im  2.2775e-8+0.0im  2.33683e-8+0.0im  2.39731e-8+0.0im  2.45895e-8+0.0im  2.52178e-8+0.0im  2.58581e-8+0.0im  2.65104e-8+0.0im  2.7175e-8+0.0im  2.7852e-8+0.0im  2.85416e-8+0.0im  2.9244e-8+0.0im  2.99592e-8+0.0im  3.06874e-8+0.0im  3.14288e-8+0.0im  3.21836e-8+0.0im  3.29519e-8+0.0im  3.37338e-8+0.0im  3.45296e-8+0.0im  3.53394e-8+0.0im  3.61633e-8+0.0im  3.70016e-8+0.0im  3.78543e-8+0.0im  3.87217e-8+0.0im  3.96039e-8+0.0im  4.05011e-8+0.0im  4.14134e-8+0.0im  4.23411e-8+0.0im  4.32842e-8+0.0im  4.42431e-8+0.0im  4.52177e-8+0.0im  4.62084e-8+0.0im  4.72152e-8+0.0im  4.82384e-8+0.0im  4.92782e-8+0.0im  5.03346e-8+0.0im  5.14079e-8+0.0im  5.24984e-8+0.0im  5.3606e-8+0.0im  5.47311e-8+0.0im  5.58738e-8+0.0im  5.70343e-8+0.0im  5.82127e-8+0.0im  5.94094e-8+0.0im  6.06243e-8+0.0im  6.18578e-8+0.0im  6.31101e-8+0.0im  6.43812e-8+0.0im  6.56714e-8+0.0im  6.6981e-8+0.0im  6.831e-8+0.0im  6.96586e-8+0.0im  7.10271e-8+0.0im  7.24157e-8+0.0im  7.38246e-8+0.0im  7.52539e-8+0.0im  7.67038e-8+0.0im  7.81746e-8+0.0im  7.96664e-8+0.0im  8.11794e-8+0.0im  8.27139e-8+0.0im  8.427e-8+0.0im  8.5848e-8+0.0im  8.7448e-8+0.0im  8.90703e-8+0.0im  9.0715e-8+0.0im  9.23824e-8+0.0im  9.40727e-8+0.0im  9.5786e-8+0.0im  9.75226e-8+0.0im  9.92828e-8+0.0im  1.01067e-7+0.0im  1.02874e-7+0.0im  1.04706e-7+0.0im  1.06562e-7+0.0im  1.08443e-7+0.0im  1.10349e-7+0.0im  1.12279e-7+0.0im  1.14235e-7+0.0im  1.16216e-7+0.0im  1.18223e-7+0.0im  1.20256e-7+0.0im  1.22314e-7+0.0im  1.24399e-7+0.0im  1.26511e-7+0.0im  1.28649e-7+0.0im  1.30815e-7+0.0im  1.33007e-7+0.0im  1.35227e-7+0.0im  1.37475e-7+0.0im  1.3975e-7+0.0im  1.42054e-7+0.0im  1.44386e-7+0.0im  1.46746e-7+0.0im  1.49135e-7+0.0im  1.51554e-7+0.0im  1.54001e-7+0.0im  1.56478e-7+0.0im  1.58985e-7+0.0im  1.61522e-7+0.0im  1.64089e-7+0.0im  1.66687e-7+0.0im  1.69315e-7+0.0im  1.71974e-7+0.0im  1.74664e-7+0.0im  1.77386e-7+0.0im  1.8014e-7+0.0im  1.82925e-7+0.0im  1.85742e-7+0.0im  1.88592e-7+0.0im  1.91475e-7+0.0im  1.9439e-7+0.0im  1.97339e-7+0.0im  2.00321e-7+0.0im  2.03336e-7+0.0im  2.06386e-7+0.0im  2.09469e-7+0.0im  2.12587e-7+0.0im  2.1574e-7+0.0im  2.18928e-7+0.0im  2.2215e-7+0.0im  2.25408e-7+0.0im  2.28702e-7+0.0im  2.32032e-7+0.0im  2.35398e-7+0.0im  2.388e-7+0.0im  2.4224e-7+0.0im  2.45716e-7+0.0im  2.49229e-7+0.0im  2.5278e-7+0.0im  2.56369e-7+0.0im  2.59995e-7+0.0im  2.6366e-7+0.0im  2.67364e-7+0.0im  2.71106e-7+0.0im  2.74888e-7+0.0im  2.78709e-7+0.0im  2.82569e-7+0.0im  2.8647e-7+0.0im  2.90411e-7+0.0im  2.94392e-7+0.0im  2.98414e-7+0.0im  3.02477e-7+0.0im  3.06581e-7+0.0im  3.10727e-7+0.0im  3.14915e-7+0.0im  3.19144e-7+0.0im  3.23417e-7+0.0im  3.27732e-7+0.0im  3.3209e-7+0.0im  3.36491e-7+0.0im  3.40936e-7+0.0im  3.45425e-7+0.0im  3.49958e-7+0.0im  3.54535e-7+0.0im  3.59157e-7+0.0im  3.63824e-7+0.0im  3.68536e-7+0.0im  3.73294e-7+0.0im  3.78097e-7+0.0im  3.82947e-7+0.0im  3.87844e-7+0.0im  3.92787e-7+0.0im  3.97777e-7+0.0im  4.02814e-7+0.0im  4.07899e-7+0.0im  4.13032e-7+0.0im  4.18214e-7+0.0im  4.23443e-7+0.0im  4.28722e-7+0.0im  4.3405e-7+0.0im  4.39427e-7+0.0im  4.44854e-7+0.0im  4.50331e-7+0.0im  4.55859e-7+0.0im  4.61437e-7+0.0im  4.67066e-7+0.0im  4.72746e-7+0.0im  4.78478e-7+0.0im  4.84262e-7+0.0im  4.90098e-7+0.0im  4.95986e-7+0.0im  5.01928e-7+0.0im  5.07922e-7+0.0im  5.1397e-7+0.0im  5.20072e-7+0.0im  5.26228e-7+0.0im  5.32438e-7+0.0im  5.38704e-7+0.0im  5.45024e-7+0.0im  5.51399e-7+0.0im  5.5783e-7+0.0im  5.64318e-7+0.0im  5.70861e-7+0.0im  5.77461e-7+0.0im  5.84119e-7+0.0im  5.90833e-7+0.0im  5.97605e-7+0.0im  6.04435e-7+0.0im  6.11324e-7+0.0im  6.18271e-7+0.0im  6.25277e-7+0.0im  6.32342e-7+0.0im  6.39467e-7+0.0im  6.46652e-7+0.0im  6.53897e-7+0.0im  6.61203e-7+0.0im  6.6857e-7+0.0im  6.75998e-7+0.0im  6.83488e-7+0.0im  6.9104e-7+0.0im  6.98654e-7+0.0im  7.0633e-7+0.0im  7.1407e-7+0.0im  7.21873e-7+0.0im  7.2974e-7+0.0im  7.3767e-7+0.0im  7.45665e-7+0.0im  7.53725e-7+0.0im  7.6185e-7+0.0im  7.7004e-7+0.0im  7.78296e-7+0.0im  7.86619e-7+0.0im  7.95007e-7+0.0im  8.03463e-7+0.0im  8.11985e-7+0.0im  8.20575e-7+0.0im  8.29233e-7+0.0im  8.37959e-7+0.0im  8.46754e-7+0.0im  8.55618e-7+0.0im  8.64551e-7+0.0im  8.73553e-7+0.0im  8.82626e-7+0.0im  8.91769e-7+0.0im  9.00983e-7+0.0im  9.10267e-7+0.0im  9.19624e-7+0.0im  9.29052e-7+0.0im  9.38552e-7+0.0im  9.48125e-7+0.0im  9.57771e-7+0.0im  9.6749e-7+0.0im  9.77282e-7+0.0im  9.87149e-7+0.0im  9.9709e-7+0.0im  1.00711e-6+0.0im  1.0172e-6+0.0im  1.02736e-6+0.0im  1.03761e-6+0.0im  1.04792e-6+0.0im  1.05832e-6+0.0im  1.06879e-6+0.0im  1.07934e-6+0.0im  1.08997e-6+0.0im  1.10068e-6+0.0im  1.11146e-6+0.0im  1.12232e-6+0.0im  1.13326e-6+0.0im  1.14429e-6+0.0im  1.15539e-6+0.0im  1.16657e-6+0.0im  1.17783e-6+0.0im  1.18918e-6+0.0im  1.2006e-6+0.0im  1.21211e-6+0.0im  1.2237e-6+0.0im  1.23538e-6+0.0im  1.24713e-6+0.0im  1.25897e-6+0.0im  1.2709e-6+0.0im  1.2829e-6+0.0im  1.295e-6+0.0im  1.30718e-6+0.0im  1.31944e-6+0.0im  1.33179e-6+0.0im  1.34422e-6+0.0im  1.35675e-6+0.0im  1.36936e-6+0.0im  1.38205e-6+0.0im  1.39484e-6+0.0im  1.40771e-6+0.0im  1.42067e-6+0.0im  1.43373e-6+0.0im  1.44687e-6+0.0im  1.4601e-6+0.0im  1.47342e-6+0.0im  1.48683e-6+0.0im  1.50033e-6+0.0im  1.51393e-6+0.0im  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Finally, we plot the time-dependent dynamics of Rydberg density for each site

clocks = [t for t in 0:1e-5:total_time]
heatmap(clocks, 1:9, D'; axis=(xlabel="iterations", ylabel="rydberg density per site"))

We can clearly see that a $Z_2$ ordered state has been generated by following the adiabatic pulse sequence. We can also confirm this by plotting out the bitstring distribution at the final time step

bitstring_histgram(reg; nlargest=20)

To prepare $Z_3$ ($Z_4$) states, we can reduce the seperation between nearby atoms to 3.57 $\mu m$ (2.87 $\mu m$).

Run in the blockade subspace

In the above example, we have run the emulation in fullspace, which can be slow and only works in small system. We can turn the above simulation into a blockade subspace simulation by changing the register to a RydbergReg by feeding a subspace object

The subspace can be found by looking up the independent set of the graph constructed by a blockade radius, here we choose the radius to be 6.2 $\mu m$

space = blockade_subspace(atoms, 6.2)

9-qubits 89-elements Subspace{Vector{Int64}}:
───┬────
  1│ 0
  2│ 1
  3│ 2
  4│ 4
  5│ 5
  6│ 8
  7│ 9
  8│ 10
  9│ 16
 10│ 17
 11│ 18
 12│ 20
 13│ 21
 14│ 32
 15│ 33
 16│ 34
 17│ 36
 18│ 37
 19│ 40
 20│ 41
 21│ 42
 22│ 64
 23│ 65
 24│ 66
 25│ 68
 26│ 69
 27│ 72
 28│ 73
 29│ 74
 30│ 80
 31│ 81
 32│ 82
 33│ 84
 34│ 85
 35│ 128
 36│ 129
 37│ 130
 38│ 132
 39│ 133
 40│ 136
 41│ 137
 42│ 138
 43│ 144
 44│ 145
 45│ 146
 46│ 148
 47│ 149
 48│ 160
 49│ 161
 50│ 162
 51│ 164
 52│ 165
 53│ 168
 54│ 169
 55│ 170
 56│ 256
 57│ 257
 58│ 258
 59│ 260
 60│ 261
 61│ 264
 62│ 265
 63│ 266
 64│ 272
 65│ 273
 66│ 274
 67│ 276
 68│ 277
 69│ 288
 70│ 289
 71│ 290
 72│ 292
 73│ 293
 74│ 296
 75│ 297
 76│ 298
 77│ 320
 78│ 321
 79│ 322
 80│ 324
 81│ 325
 82│ 328
 83│ 329
 84│ 330
 85│ 336
 86│ 337
 87│ 338
 88│ 340
 89│ 341

Then we create our register in subspace

reg = zero_state(space)

RydbergReg{ComplexLayout, Vector{ComplexF64}, Subspace{Vector{Int64}}}(9, ComplexLayout(), ComplexF64[1.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im], Subspace{Vector{Int64}}(9, OrderedCollections.OrderedDict(0 => 1, 1 => 2, 2 => 3, 4 => 4, 5 => 5, 8 => 6, 9 => 7, 10 => 8, 16 => 9, 17 => 10, 18 => 11, 20 => 12, 21 => 13, 32 => 14, 33 => 15, 34 => 16, 36 => 17, 37 => 18, 40 => 19, 41 => 20, 42 => 21, 64 => 22, 65 => 23, 66 => 24, 68 => 25, 69 => 26, 72 => 27, 73 => 28, 74 => 29, 80 => 30, 81 => 31, 82 => 32, 84 => 33, 85 => 34, 128 => 35, 129 => 36, 130 => 37, 132 => 38, 133 => 39, 136 => 40, 137 => 41, 138 => 42, 144 => 43, 145 => 44, 146 => 45, 148 => 46, 149 => 47, 160 => 48, 161 => 49, 162 => 50, 164 => 51, 165 => 52, 168 => 53, 169 => 54, 170 => 55, 256 => 56, 257 => 57, 258 => 58, 260 => 59, 261 => 60, 264 => 61, 265 => 62, 266 => 63, 272 => 64, 273 => 65, 274 => 66, 276 => 67, 277 => 68, 288 => 69, 289 => 70, 290 => 71, 292 => 72, 293 => 73, 296 => 74, 297 => 75, 298 => 76, 320 => 77, 321 => 78, 322 => 79, 324 => 80, 325 => 81, 328 => 82, 329 => 83, 330 => 84, 336 => 85, 337 => 86, 338 => 87, 340 => 88, 341 => 89), [0, 1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, 21, 32, 33, 34, 36, 37, 40, 41, 42, 64, 65, 66, 68, 69, 72, 73, 74, 80, 81, 82, 84, 85, 128, 129, 130, 132, 133, 136, 137, 138, 144, 145, 146, 148, 149, 160, 161, 162, 164, 165, 168, 169, 170, 256, 257, 258, 260, 261, 264, 265, 266, 272, 273, 274, 276, 277, 288, 289, 290, 292, 293, 296, 297, 298, 320, 321, 322, 324, 325, 328, 329, 330, 336, 337, 338, 340, 341]))

The rest of code will be the same as the full space

prob = SchrodingerProblem(reg, total_time, h)
emulate!(prob)
bitstring_histgram(prob.reg; nlargest=20)

State preparation in 2D

Now we will show how to prepare a 2D checkboard phase. Most of codes will be the same as the 1D case, except that we will choose slightly different parameters and specify a square lattice

nx, ny = 3, 3
nsites = nx*ny
atoms = generate_sites(SquareLattice(), nx, ny, scale=6.7)

total_time = 2.9
Ω_max = 2π * 4.3
Ω = piecewise_linear(clocks=[0.0, 0.3, 2.6, total_time], values=[0.0, Ω_max , Ω_max , 0])
draw(Ω)

U = 2π * 15.0
Δ = piecewise_linear(clocks=[0.0, 0.3, 2.6, total_time], values=[-U, -U, U , U])
draw(Δ)

h = rydberg_h(atoms; Δ, Ω)


reg = zero_state(9)
prob = SchrodingerProblem(reg, total_time, h)
integrator = init(prob, Vern8())
densities = []

Any[]

If you are using adaptive steps (by default), you can use TimeChoiceIterator to specify the points you would like to stop by

for _ in TimeChoiceIterator(integrator, 0.0:1e-3:total_time)
    push!(densities, [expect(put(nsites, i=>Op.n), reg) for i in 1:nsites])
end
D = hcat(densities...)

clocks = [t for t in 0:1e-3:total_time]
heatmap(clocks, 1:9, D'; axis=(xlabel="iterations", ylabel="rydberg density per site"))

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